Question

which of the following represents a geometric sequence?
i \t\t\t$\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \dots$
ii \t\t\t$\frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \dots$
iii \t\t\t$\frac{1}{4}, 1, -4, 16, \dots$
iv \t\t\t$\frac{1}{4}, -4, \frac{1}{4}, -4, \dots$
\bigcirc i
\bigcirc ii
\bigcirc iii
\bigcirc iv

Explanation:

Step1: Recall Geometric Sequence Definition

A geometric sequence has a common ratio \( r \), where \( r=\frac{a_{n + 1}}{a_n} \) for all \( n \).

Step2: Analyze Sequence I

Sequence I: \( \frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4},\dots \)
Calculate \( r \): \( \frac{\frac{1}{4}}{\frac{1}{4}} = 1 \), \( \frac{\frac{1}{4}}{\frac{1}{4}} = 1 \), etc. Common ratio \( r = 1 \), so it is a geometric sequence. But let's check others.

Step3: Analyze Sequence II

Sequence II: \( \frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{1}{7},\dots \)
Calculate \( r \): \( \frac{\frac{1}{5}}{\frac{1}{4}}=\frac{4}{5} \), \( \frac{\frac{1}{6}}{\frac{1}{5}}=\frac{5}{6} \). Ratios are not equal, so not geometric.

Step4: Analyze Sequence III

Sequence III: \( \frac{1}{4},1, - 4,16,\dots \)
Calculate \( r \): \( \frac{1}{\frac{1}{4}} = 4 \), \( \frac{-4}{1}=-4 \). Ratios \( 4
eq - 4 \), so not geometric.

Step5: Analyze Sequence IV

Sequence IV: \( \frac{1}{4},-4,\frac{1}{4},-4,\dots \)
Calculate \( r \): \( \frac{-4}{\frac{1}{4}}=-16 \), \( \frac{\frac{1}{4}}{-4}=-\frac{1}{16} \). Ratios \( - 16
eq-\frac{1}{16} \), so not geometric. Wait, but Sequence I has \( r = 1 \), but let's re - check. Wait, maybe I made a mistake. Wait, Sequence I: all terms same, ratio 1. But let's check the options again. Wait, maybe the original problem's Sequence I is \( \frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}\dots \), ratio 1. But let's check the options. Wait, the options are I, II, III, IV. Wait, maybe I misread Sequence IV. Wait, Sequence IV: \( \frac{1}{4}, - 4,\frac{1}{4}, - 4,\dots \). Wait, \( \frac{-4}{\frac{1}{4}}=-16 \), \( \frac{\frac{1}{4}}{-4}=-\frac{1}{16} \), not equal. Sequence III: \( \frac{1}{\frac{1}{4}} = 4 \), \( \frac{-4}{1}=-4 \), not equal. Sequence II: not geometric. Sequence I: ratio 1, so it is geometric. But wait, maybe the problem's Sequence I is correct. Wait, the options: the first option is I. But let's confirm.

Wait, another way: A geometric sequence can have ratio 1 (constant sequence). So Sequence I: each term is \( \frac{1}{4} \), so \( a_{n+1}=a_n\times1 \), so it is geometric. Sequence II: harmonic? No, geometric ratio changes. Sequence III: ratio changes. Sequence IV: ratio changes. So the correct one is I.

Answer:

I. \( \frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4},\dots \)